| L(s) = 1 | − 9·3-s + 67.2·5-s + 49·7-s + 81·9-s + 17.2·11-s + 807.·13-s − 604.·15-s + 777.·17-s − 262.·19-s − 441·21-s + 1.95e3·23-s + 1.39e3·25-s − 729·27-s − 1.51e3·29-s − 5.47e3·31-s − 154.·33-s + 3.29e3·35-s + 6.58e3·37-s − 7.26e3·39-s + 2.93e3·41-s + 975.·43-s + 5.44e3·45-s − 2.29e4·47-s + 2.40e3·49-s − 6.99e3·51-s + 2.64e3·53-s + 1.15e3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.20·5-s + 0.377·7-s + 0.333·9-s + 0.0428·11-s + 1.32·13-s − 0.694·15-s + 0.652·17-s − 0.166·19-s − 0.218·21-s + 0.768·23-s + 0.445·25-s − 0.192·27-s − 0.335·29-s − 1.02·31-s − 0.0247·33-s + 0.454·35-s + 0.791·37-s − 0.765·39-s + 0.272·41-s + 0.0804·43-s + 0.400·45-s − 1.51·47-s + 0.142·49-s − 0.376·51-s + 0.129·53-s + 0.0515·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.562367934\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.562367934\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 7 | \( 1 - 49T \) |
| good | 5 | \( 1 - 67.2T + 3.12e3T^{2} \) |
| 11 | \( 1 - 17.2T + 1.61e5T^{2} \) |
| 13 | \( 1 - 807.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 777.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 262.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.95e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.51e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.47e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.58e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.93e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 975.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.29e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.64e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.25e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 9.53e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.78e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.01e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.01e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.77e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.34e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.20e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.86e4T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.81981832218143033881872054091, −9.832843096974503199850946220006, −9.025965551373310371264767673565, −7.88128231372086693473316197611, −6.59117506155970808072504152234, −5.83978320655420270493131499026, −5.01366529594419898732142427779, −3.57376405158692377976956174197, −1.97616531283668705982532057672, −0.970734608851987316681742696385,
0.970734608851987316681742696385, 1.97616531283668705982532057672, 3.57376405158692377976956174197, 5.01366529594419898732142427779, 5.83978320655420270493131499026, 6.59117506155970808072504152234, 7.88128231372086693473316197611, 9.025965551373310371264767673565, 9.832843096974503199850946220006, 10.81981832218143033881872054091
